Constraints on the Redshift Evolution of the L_X-SFR Relation from the Cosmic X-Ray Backgrounds
aa r X i v : . [ a s t r o - ph . C O ] N ov Mon. Not. R. Astron. Soc. , 1–12 (2009) Printed 13 November 2018 (MN LaTEX style file v2.2)
Constraints on the Redshift Evolution of the L X -SFRRelation from the Cosmic X-Ray Backgrounds Mark Dijkstra ⋆ , Marat Gilfanov , , Abraham Loeb , and Rashid Sunyaev , Max-Planck Institut fuer Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany Space Research Institute of Russian Academy of Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia Astronomy Department, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA
13 November 2018
ABSTRACT
Observations of local star forming galaxies have revealed a correlation between the rateat which galaxies form stars and their X-Ray luminosity. We combine this correlationwith the most recent observational constraints on the integrated star formation ratedensity, and find that star forming galaxies account for 5-20% of the total soft andhard X-ray backgrounds, where the precise number depends on the energy band andthe assumed average X-ray spectral energy distribution of the galaxies below ∼ L X -SFR relation with recently derived star formation ratefunction, then we find that star forming galaxies whose X-ray flux falls well (morethan a factor of 10) below the detection thresholds of the Chandra Deep Fields, canfully account for the unresolved soft X-ray background, which corresponds to ∼ c X ≡ L X /SFR, and/or its evolution towards lower and higher starformation rates. If we parametrize the redshift evolution of c X ∝ (1 + z ) b , then we findthat b . c X may be increasing towards higherredshifts and/or higher star formation rates at levels that are consistent with theX-ray background, but possibly at odds with the locally observed L X -SFR relation. Key words: galaxies: high redshift – galaxies: stellar content – X-rays: binaries –X-rays: galaxies
Galaxies contain various sources of X-ray emission which in-clude: ( i ) active galactic nuclei (AGN), which are poweredby accretion of gas onto a supermassive black hole ( ii ) hot( T > ∼ K) interstellar gas, and ( iii ) X-ray binaries, whichconsist of a compact object, either a neutron star or a stellarmass black hole, and a companion star from which the com-pact object accretes mass. The X-Ray luminosity of starforming galaxies without AGN is dominated by so-calledhigh-mass X-ray binaries (HMXBs, e.g. Grimm et al. 2003),in which the neutron star or stellar mass black hole is accret-ing gas from a companion that is more massive than ∼ M ⊙ .HMXBs are thus tightly linked to massive stars, and sincemassive stars are short lived, the combined X-ray luminosityof HMXBs is expected to be linked to the rate at which starsform (e.g. Helfand & Moran 2001). The X-Ray luminosityof star forming galaxies is indeed observed to be correlatedwith the rate at which they are forming stars (Grimm et al. ⋆ E-mail:[email protected] L X -SFR relation’ encodes a wealth of informa-tion on various astrophysical processes. These include theinitial mass function (IMF) of stars, the fraction of mas-sive stars that form in binaries, the mass ratio distributionof binary stars, the distribution of their separations, thegas metallicity, and the common envelope efficiency (e.g.Belczynski et al. 2002; Mineo et al. 2011b, and referencestherein). Despite its dependence on many astrophysical pro-cesses, the L X -SFR relation is observed to hold over ∼ σ = 0 . L X -SFR relation in nearby galaxies. There are observationalhints–as well as theoretical expectations– that c X is higherin low mass galaxies and/or low metallicity environments(Dray 2006; Linden et al. 2010; Kaaret et al. 2011), whichsuggests that c X ≡ L X / SFR could have been substantiallyhigher at higher redshifts (see Mirabel et al. 2011, for a c (cid:13) Dijkstra et al. summary). However, quantitative constraints on ratio c X athigher redshifts are virtually non-existent. The main reasonis that the X-ray flux that reaches Earth from individualstar forming galaxies typically falls well below the detectionthreshold of existing X-ray telescopes (see § L X also plays animportant role in determining the thermal history ofintergalactic medium at very high redshifts (e.g. Oh2001; Venkatesan et al. 2001), and strongly affects the21-cm signal from atomic hydrogen during the darkages (Furlanetto et al. 2006; Pritchard & Furlanetto 2007;Pritchard & Loeb 2010; Alvarez et al. 2010). Current theo-retical models of this reheating process explore values for c X that span ∼ c x witheither redshift and/or towards high/low star formation ratesthan probed by existing observations of individual galaxies,using the cosmic X-Ray backgrounds (CXBs).Our paper is organized as follows. In §
2, we summarizerecent observational constraints on the levels of the total X-ray background (XRB, both soft and hard), as well as thefraction of the XRB that has been resolved in discrete X-ray sources. In § § § L X -SFR relation using the unresolvedsoft XRB. There are several additional candidate sourceswhich contribute to the unresolved CXBs (these include forexample low mass X-ray binaries and weak AGN, see § c X we will allow the entire un-resolved SXB to be produced by HMXBs (and the hot ISM)in X-ray faint star-forming galaxies. This results in conserva-tive upper limits on possible evolution in c X . Finally, we con-clude in §
6. The cosmological parameters used throughoutour discussion are (Ω m , Ω Λ , Ω b , h ) = (0 . , . , . , . The total soft ( E =1-2 keV) CXB (SXB) amounts to S − = 4 . ± . × − erg s − cm − deg − (e.g.Hickox & Markevitch 2006). Hickox & Markevitch (2006)find an unresolved SXB intensity of 1 . ± . × − erg s − cm − deg − , after removing all point and extendedsources detected in the Chandra Deep Fields (CDFs). Thedetection threshold in the
Chandra Deep Field-North (CDF-North) corresponds to s th ∼ . × − erg s − cm − (Alexander et al. 2003; Hickox & Markevitch 2007a).Hickox & Markevitch (2007a) showed through a stack-ing analysis that ∼
70% of the unresolved component isaccounted for by sources that are detected with the
HubbleSpace Telescope , but not individually as X-ray sources. Theirstacking analysis shows that these X-ray undetected sourceshave an average
X-ray flux that is h s i ∼ . − . s th ∼ . − . × − erg s − cm − . Hickox & Markevitch (2007b)find that the cumulative number of these X-ray unde-tected HST sources brighter than some of X-ray flux s ,is well-described by a power-law N ( > s ) ∝ s − β , where β = 1 . +0 . − . . We can then estimate the minimum X-rayflux, s min , that the HST-detected sources probed from h s i = R s th s min dS dNdS S/ h R s th s min dS dNdS i , and find that s min =1 . − . × − erg s − cm − (for β = 1 . . ± . × − erg s − cm − deg − (Hickox & Markevitch2007a), down to a minimum flux s min . The total hard ( E =2-8 keV) CXB (HXB) amounts to S − = 1 . ± . × − erg s − cm − deg − (e.g.Hickox & Markevitch 2006). Hickox & Markevitch (2006)find an unresolved HXB intensity of 3 . ± . × − ergs − cm − deg − . We will not use the unresolved HXB to putconstraints on the redshift evolution of c X for two reasons:( i ) the uncertainties on the unresolved HXB are larger thanfor the SXB, and ( ii ) as we will explain below ( § The total contribution S tot (in erg s − cm − deg − ) of starforming galaxies to the SXB is given by (Appendix B) S tot = ∆Ω4 π cH Z z max dz (1 + z ) E ( z ) ˙ ρ ∗ ( z ) L X ( z, Γ) . (1)Here ∆Ω ∼ . × − sr deg − , E ( z ) = p Ω m (1 + z ) + Ω Λ ,and ˙ ρ ∗ ( z ) denotes the comoving star formation rate densityat redshift z (in M ⊙ yr − cMpc − , where ‘cMpc’ standsfor co-moving Mpc). We adopt the star formation historyfrom Hopkins & Beacom (2006), using the parametric form˙ ρ ∗ ( z ) = ( a + bz ) h/ (1+( z/c ) d ) from Cole et al. (2001), where a = 0 . b = 0 . c = 3 .
3, and d = 5 .
3. We note that theamplitude of the function ˙ ρ ∗ ( z ) depends on the assumedIMF (see Hopkins & Beacom 2006, for a more detailed dis-cussion). The adopted normalization derives from classi-cal Salpeter IMF (Salpeter 1955) between 0.1 to 100 M ⊙ (Hopkins 2004). The same IMF was assumed in the deriva-tion of the SFR- L X relation (see Mineo et al. 2011b), andour calculations are therefore self-consistent.The term L ( z, Γ) ≡ c X K ( z, Γ) denotes the ‘K-corrected’X-ray luminosity (in erg s − ) per unit star formation ratein the observed energy range E -E . In this paper, E = 1 . = 2 . = 2 . = 8 . the value of c X ≡ L X /SFR, The literature contains values for c X that at face value ap-c (cid:13) , 1–12 RB Constraints on the SFR-L X Relation Figure 1.
The top panels show the fraction of the total soft ( left ) and hard ( right ) X-Ray backgrounds that can be attributed to starforming galaxies, as a function of the assumed photon index Γ, for power-law X-Ray spectral energy distributions (see text). We havedrawn curves for a range of observed values for c X ≡ L X / SFR (see text). Depending on the assumed value for Γ, star forming galaxiescan account for ∼ −
15% of the total observed soft X-ray background, and up to ∼
20% of the hard X-ray background. For comparison,Swartz et al. (2004) found that typically ULX spectra in the Chandra bands could be described by a powerlaw with h Γ i = 1 .
7, whichwould place the contribution of star forming galaxies to the soft X-ray background at ∼ − E > lower panels showthat the contribution per unit redshift, dS tot /dz , peaks at low redshift z ∼ . − . where SFR denotes the star formation rate in M ⊙ yr − , tobe c X = 2 . × erg s − / [ M ⊙ yr − ] when only compact re-solved X-Ray sources in galaxies are included. Mineo et al.(2011b) also found that the best fit c X , max = 3 . × erg s − [ M ⊙ yr − ] − for unresolved galaxies in the ChandraDeep Field North and ULIRGs. However, a non-negligiblefraction of this additional unresolved flux is in a soft com-ponent, and would not contribute to soft X-Ray background(measured in the 1-2 keV band, see Bogdan & Gilfanov2011). We will not attempt to model in detail the contribu-tion of unresolved X-Ray emission. Instead, Figures 1, 2, and pear both lower (e.g. Persic & Rephaeli 2007) and higher (e.g.Ranalli et al. 2003) by a factor of a few. However, some studiesmeasured the X-Ray luminosity ( L X ) in the range 2-10 keV (e.g.Gilfanov et al. 2004; Persic et al. 2004; Persic & Rephaeli 2007;Lehmer et al. 2010), or 0.5-2.0 keV (Ranalli et al. 2003). Further-more, some studies have derived values for c X using a forma-tion rate of stars (SFR) in the mass range 5 M/M ⊙ c X .Mineo et al. (2011b) discuss that most studies are consistent withtheir derived value when identical definitions for ‘SFR’ and ‘ L X ’are used. c x = 2 . − . × ergs − [ M ⊙ yr − ] − . We point out that Mineo et al. (2011b)measured the X-ray luminosity over the energy range 0 . − L X and SFR holds over ∼ ∼ . − M ⊙ yr − , with a scatter of σ ∼ . c X with a mean of h log c X i = 39 . σ = 0 . c X , it is skewed towards larger values of c X in linear coordinates. The observed scatter in the L X -SFRrelation therefore enhances our computed contributions tothe X-ray backgrounds. In the absence of this scatter, S tot reduces by a factor of exp (cid:0) − σ ln (cid:1) ∼ . z to the CXB in the observed energy range E -E , we need to compute the galaxy’s luminosity in therange [E -E ] × (1 + z ) keV. In analogy to the standard‘K-correction’ (e.g. Hogg et al. 2002), we multiply c X by K X ( z, Γ) = I ( E (1 + z ) , E (1 + z )) / I (0 . , . I ( x, y ) = R yx En ( E ) dE , where n ( E ) dE denotes the c (cid:13) , 1–12 Dijkstra et al. number of emitted of photons in the energy range E ± dE/ ψ . In this paper, we explore powerlaw SEDs for the form n ( E ) ∝ E − Γ , and the integrals I ( x, y )can be evaluated analytically. Finally, we take the redshiftintegral from z min = 0 to z max = 10. The results are onlyweakly dependent on the integration limits we pick, as longas z max > . lower right panel of Fig. 2). The top panels of Figure 1 show the fraction of the totalsoft ( left panel , 1-2 keV) and hard ( right panel , 2-8 keV)X-Ray backgrounds that can be attributed to star forminggalaxies, as a function of the assumed photon index Γ, andfor a realistic range of c X (see above). Depending on theassumed value for Γ, star forming galaxies can account for ∼ −
15% of the total soft X-ray background, and up to ∼ h Γ i =1 .
7, and mode (i.e. most likely value) of Γ pk ∼ . . − . ∼ − E >
10 keV.The lower panels show the differential contribution asa function of redshift. This differential contribution is givenby dS tot /dz . These plots show that dS tot /dz peaks at z ∼ .
38, and that the dominant contribution to the total X-Ray background comes from lower redshifts. Indeed, ∼ ∼ z < ∼ . z < ∼ . § We can compute the contribution S X of star forming galax-ies, fainter than some observed soft X-ray flux s max , to theSXB as S X = ∆Ω4 π cH Z z max dz (1 + z ) E ( z ) × (2) Z L X , max d log L X n (log L X , z ) L X K X ( z, Γ) , where n (log L X , z ) d log L X denotes the comoving numberdensity of star forming galaxies with X-ray luminosities inthe range log L X ± d log L x / n (log L X , z ) arecMpc − dex − ). Here, L X denotes the X-ray luminosity ofgalaxies in the 0.5-8.0 keV (restframe). The integral over L X then extends up to L X , max ≡ πd ( z ) s max /K X ( z, Γ), where d L ( z ) is the luminosity distance to redshift z . The quantity n (log L X ), also referred to as the X-ray luminosity function(XLF) of star forming galaxies, is given by n (log L X , z ) = Z ψ max ψ min dψ n ( ψ, z ) P (log L X | ψ ) , (3) where n ( ψ, z ) dψ denotes ’star formation rate function’,which gives the comoving number density of galaxies thatare forming stars at a rate SFR= ψ ± dψ/ z . Thefunction P (log L X | ψ ) d log L x denotes the probability that agalaxy that is forming stars at a rate ψ has an X-ray lu-minosity in the range log L X ± d log L x /
2. We describe bothfunctions in more details below. We start the integral over ψ at ψ min = 10 − M ⊙ yr − , which corresponds approximatelyto the SFR that is theoretically expected to occur in darkmatter halos of mass M halo ∼ M ⊙ (Wise & Cen 2009;Trac & Cen 2007; Zheng et al. 2010). Our final results de-pend only weakly on ψ min (see the upper left panel of Fig. 2).The ψ -integral extends up to ψ max = 10 M ⊙ yr − , with ourresults being insensitive to this choice.In the local Universe ( z ∼ n ( ψ, z ) [unitsare cMpc − ( M ⊙ yr − ) − ] appears to be described accu-rately by a Schechter function (Bothwell et al. 2011) n ( ψ, z ) = Φ ∗ ψ ∗ (cid:16) ψψ ∗ (cid:17) α e − ψ/ψ ∗ , (4)where α = − . ± .
08, Φ ∗ = (1 . ± . × − cMpc − ,and ψ ∗ = 9 . ± . M ⊙ yr − . The redshift evolution of n ( ψ, z ) is not well known. We assume throughout that α ( z ) = − . − . g ( z ), where g ( x ) ≡ π arctan x is a func-tion that obeys g (0) = 0 and lim z →∞ g ( z ) = 1. This steepeningof the low-end of the star formation rate function at higherredshifts reflects the steepening of UV luminosity functionstowards higher redshifts (e.g. Arnouts et al. 2005, Reddy& Steidel 2009, Bouwens et al. 2006, 2007,2008), and theobservation that dust-obscuration is negligible for the UV-faint galaxies (e.g. Bouwens et al. 2009). The factor ‘-0.23’causes α → − .
74 at high redshift, which corresponds tothe best-fit slope of the UV-luminosity function of z = 6drop-out galaxies (Bouwens et al. 2007). The redshift evolu-tion of Φ ∗ and ψ ∗ is more difficult to infer from the redshiftevolution of the UV luminosity functions, because of dust.We have taken two approaches: we constrain either the red-shift evolution of Φ ∗ or ψ ∗ -while keeping the other fixed- tomatch the inferred redshift evolution of the star formationrate density (see Appendix A for more details). In reality,we expect both parameters to evolve with redshift, and thatour two models bracket the range of plausible more realisticmodels.The function P (log L X | ψ ) is given by a lognormal dis-tribution P (log L X | ψ ) = 1 √ πσ exp h − (cid:0) log L X h L X i (cid:1) σ i , (5)where h L X i = c X × ψ denotes the X-ray luminosity (mea-sured in the 0.5-8.0 keV rest frame) that is expected from theobserved SFR- L X relation. The standard deviation σ = 0 . Previous work showed that this star formation rate functioncan be described by a log-normal function (Martin et al. 2005).However, this lognormal function does not provide a good fit tothe observed star formation rate function particularly for low andhigh star formation rates (see Fig 4 of Bothwell et al. 2011).c (cid:13) , 1–12
RB Constraints on the SFR-L X Relation Figure 2.
The contribution S X to the soft X-Ray background (SXB, E =1-2 keV in the observer’s frame) by galaxies whose individualsoft X-ray flux is less than s max . The median unresolved SXB is represented by the black solid horizontal lines , and its 68% confidencelevels by the the gray region , bounded by black dashed lines (taken from Hickox & Markevitch 2007a),. The red solid lines show S X as afunction of various model parameters. Our fiducial model assumes Γ = 2 . s max = 2 . × − erg s − cm − , ψ min = 0 . M ⊙ yr − , and z max = 10. The upper left panel shows that ψ min only weakly affects S X , because the faint end of the star formation function (especiallyat low z ) is not steep. The upper right panel shows that S X depends weakly on Γ, provided that Γ < ∼
2. The lower left panel shows that S x also depends weakly on s max , unless s max < ∼ − erg s − cm − . The dotted vertical line shows the X-ray detection threshold in CDF-N(Alexander et al. 2003). The black filled circle on this line shows the unresolved SXB derived by Hickox & Markevitch (2006, i.e. beforesubtracting the contribution from X-Ray faint HST detected sources). The gray region here brackets the effective minimum X-ray flux s min that is probed by stacking X-ray undetected HST sources (see § lower right panel shows that S X again depends weaklyon z max , provided that z max >
4. See the main text for a more detailed interpretation of these plots. These plots show that star forminggalaxies that are too faint to be detected as individual X-ray sources, can account for the full unresolved SXB , and that this statementis insensitive to details in the model when Γ < ∼
2, which is reasonable given the available observational constraints (Swartz et al. 2004). assumed a lognormal conditional probability functions for P (log L X | L Y ), where L Y denotes the galaxy luminosity insome other band Y. Our fiducial model assumes Γ = 2 . pk , see above), s max = 2 . × − erg s − cm − (close tothe middle of the range for s min that was quoted in § ψ min = 10 − M ⊙ yr − , and z max = 10. As mentioned pre-viously, we explore two choices for extrapolating the starformation rate function with redshift, which likely bracketthe range of physically plausible models. When we evolve ψ ∗ ( z ), but keep Φ ∗ fixed, we find that our fiducial modelgives S X = 2 . × − erg s − deg − cm − . On the otherhand, when we evolve Φ ∗ ( z ), but keep ψ ∗ fixed, we obtain S X = 2 . × − erg s − deg − cm − . The fact that thisdifference is small is encouraging, and suggests that our ig-norance of the star formation rate function at z > S X . To be conservative, we focus on thefirst model in the reminder of this paper.Figure 2 has four panels, each of which shows the me-dian unresolved soft X-ray background ( black solid lines ),and its 68% confidence levels (the gray region , bounded by black dashed lines ) from Hickox & Markevich (2007b). The red solid bands show the contribution from galaxies whoseindividual soft X-ray flux (1-2 keV observed frame) is lessthan s max . • In the upper left panel we plot S X as a function of ψ min . We find that S X depends only weakly on ψ min . Thatis, very faint galaxies do not contribute significantly to S X . This is because the majority of the contribution to S X comes from galaxies at z < ∝ ψ − . , and the overall star formation rate density is domi- c (cid:13) , 1–12 Dijkstra et al. nated by galaxies that are forming stars at a rate close to ψ ∗ . • The upper right panel shows S X as a function of Γ.We find that S X decreases with Γ. As most of the contri-bution to S X comes from galaxies at z <
2, we are mostsensitive to the X-ray emissivity of star forming galaxies at E = [1 − × (1 + z ) < [3 −
6] keV (restframe). For fixed L X , increasing Γ reduces the fraction of the emitted fluxat these ‘higher’ energies for steeper spectra (i.e. most ofthe energy lies near E = 0 . S X . • The lower left panel shows S X as a function of s max .The vertical dotted line shows the detection thresholdin the Chandra Deep Field-North (Alexander et al. 2003;Hickox & Markevitch 2007a). The gray region here bracketsthe effective minimum X-ray flux s min that is probed bystacking X-ray undetected HST sources (see § S x depends only weakly on s max , unless s max < ∼ − erg s − cm − . This weak dependence on s max at largerfluxes can be easily understood: most of the contributionto S X comes from z < ∼ z = 1, s max = 2 . × − erg s − cm corresponds to L X = 4 πd ( z ) s max / L X ( z, Γ) = 2 . × erg s − , whichrequires ψ = 9 . M ⊙ yr − , which is close to ψ ∗ . We aretherefore practically integrating over the full UV-luminosityfunction. Boosting s max therefore barely increases S X further. • The lower right panel shows S X as a function of z max .This plots shows that S X evolves most up to z max ∼
2, andbarely when z max > ∼
4. That is, galaxies at higher redshiftbarely contribute to S X as we also showed in Figure 1 (unlessthe conversion factor c X between L X and SFR changes withredshift, see § starforming galaxies that are too faint to be detected as individ-ual X-ray sources, can account for the full unresolved SXB ,and that this statement is insensitive to details in the modelwhen Γ < ∼ h Γ i = 1 . L X RELATION5.1 Constraints from the SXB
Since our fiducial model already saturates the unresolvedSXB, we can ask what constraints we can set on the redshift-dependence of the L X -SFR relation . Clearly, the SXB canonly put constraints on models in which c X increases with As we already stated in §
1, there are several additional candi-date sources which contribute to the unresolved SXB (see § c X we will conservativelyallow the entire unresolved SXB to be produced by HMXBs (andthe hot ISM) in X-ray faint star-forming galaxies. Obviously, thisresults in upper limits on possible evolution in c X . Figure 3.
The unresolved SXB constraints on the parameters A and b for a redshift evolution parametrization of the form c X ≡ L X SFR = A (1 + z ) b . Models that lie in the red region saturate theunresolved SXB at > σ (see § grey region denotes thevalue for c X ( z = 0) ≡ A derived by Mineo et al. (2011b) (the solid vertical line denotes their best-fit value). For this value of A , b < ∼ . σ ). redshift. We consider models for which L X SFR = c X ≡ A (1 + z ) b , (6)with b >
0, and investigate the constraints that the SXBplaces on the parameters A and b . We compute S X (seeEq. 3) on a grid of models which cover a range of A and b . Figure 3 shows how many σ (= 1 . × − erg s − cm − deg − ) above the median observed unresolved SXB, S obs = 3 . × − erg s − cm − deg − , the models lie.Models that lie above the uppermost solid line , indicatedby the label ‘3- σ ’ result in S X − S obs > σ . That is, thesemodels significantly overproduce the unresolved SXB andare practically ruled out. The light grey region bounded bythe dotted vertical lines indicates the 68% confidence regionfor the observed value for c X by Mineo et al. (2011b), wherewe assumed 0.4 dex uncertainty on c X . Their best-fit valuefor c X is indicated by the solid vertical line . Figure 3 showsthat for this value of A , b < ∼ . > ∼ σ . If we marginalize over A , then we also get b < ∼ . L X relation ofthe form c X → f × c X model I SFR x M ⊙ yr − ;model II SFR > y M ⊙ yr − models III&IV z > z c (7)The upper left panel of Figure 4 shows S X as a function of f for model I for x = 0 . red band ) and x = 0 . green band ).This plot shows that f > ∼ − x = 0 . We obtain this marginalized upper limit U ( b ) = R A max A min dA U ( b | A ) P prior ( A ). Here, U ( b | A ) denotes the upperlimit on b given A , and P prior ( A ) denotes our prior on theprobability density function for A . Since A ≡ c X ( z = 0),we took P prior ( A ) to be a lognormal distribution with h log c X i = 39 .
4, and σ = 0 . A min = 10 . erg s − [ M ⊙ yr − ] − , and A max = 10 . erg s − [ M ⊙ yr − ] − .c (cid:13) , 1–12 RB Constraints on the SFR-L X Relation Figure 4.
Same as Figure 2, but we modify our fiducial model such that the X-ray emission from star forming galaxies is boosted by afactor of c X → f × c X (i.e. L X → f × L X ) when SFR < x M ⊙ yr − ( top left panel ), SFR > y M ⊙ yr − ( top right panel ), and z > z c ( lowerleft panel ) as a function of f . The lower right panel shows a model where we boost c X by a factor of 100 ( upper blue band ), 10 ( middlered band ) and 3 ( lower green band ) at redshift z jump as a function of z jump . This figure shows that even small boosts (i.e. f ∼ a few)for SFR < . M ⊙ yr − or z > c X at the large SFRend barely affects S X . While large ’jumps’ in c X are not allowed if these occur at low redshift (i.e. z < ∼ z > unresolved SXB. This can be understood from the top leftpanel of Figure 2, which shows that adopting ψ min = 0 . M ⊙ yr − results in S X ∼ . × − erg s − cm − deg − , whichcorresponds to a reduction of ∼ < SFR M ⊙ yr − < . ∼
25% to S X (for f = 1).Boosting their contribution by a factor f > ∼ − c X ) causes S X to exceed the unresolved SXB.The upper right panel shows that the soft XRB onlyallows constraints to be put on f > ∼
10, and only if y < ∼ f are not possible for large ψ . This isbecause galaxies that are forming stars at a rate ψ > ∼ M ⊙ yr − are deep in the exponential tail of the star formationfunction. As a result of their small number density, theybarely contribute anything to S X .The lower left panel shows that boosting c X at z > f > ∼ − z > S X for our fiducial choice of c X . However, theunresolved SXB cannot place tight limits (yet) on c X at veryhigh redshifts. This is illustrated in the lower right panel where we show S x for model IV: once z jump > ∼
5, boosting c X by as much as ∼
100 has little impact on S X . As part of our analysis, we compute theoretical galaxyX-Ray luminosity functions (XLFs) using equation (3).Observed galaxy XLFs have been presented, for example,by Norman et al. (2004); Georgantopoulos et al. (2005);Georgakakis et al. (2006); Tzanavaris & Georgantopoulos(2008). Tzanavaris & Georgantopoulos (2008) presentgalaxy XLF for late-type (i.e. star forming) galaxies intwo redshifts bins: the first bin contains galaxies with0 < z < . z med = 0 . . z < . z med = 0 . data points show the observed X-Rayluminosity functions (XLFs) from Tzanavaris & Georgan-topoulos (2008), while the red dotted line shows the best-fit Schechter function derived by Georgakakis et al (2006).The solid lines show predictions for our fiducial model c X = A (1+ z ) b with A = 2 . × erg s − [ M ⊙ yr − ] − , and b = 0 .
0. At z . ∼ −
3. We obtaina better fit if we lower A = 1 . × erg s − / [ M ⊙ yr − ], c (cid:13) , 1–12 Dijkstra et al. which corresponds to ∼ . A corresponds to almost exactly the value quotedby Lehmer et al (2010, their β , although these authors mea-sured L X in the 2.0-10.0 keV range). At higher redshift,our model significantly underpredicts the number density at L X , . − . > ∼ erg s − . This may suggest that either thatthe XLFs (strongly) favor c X to increase towards higher red-shift, and/or SFR > ∼ a few tens of M ⊙ yr − (see below).Alternatively, the observed XLFs of star forming galaxies arecontaminated by low luminosity AGN, which are difficult toidentify at these X-ray luminosities.Our predicted XLFs agree quite well with previous pre-dictions by Ranalli et al. (2005), for the model in which theyassume that the redshift evolution of the luminosity func-tions is solely the result of evolution in the number densityof galaxies (this is referred to as ‘density evolution’). Theirmodel also underpredicts the number density of luminousX-ray sources at higher redshift. Ranalli et al. (2005) foundthat a better fit to the high-redshift data is obtained fora model in which solely the luminosity of galaxies evolves(‘luminosity evolution’) as ∝ (1 + z ) . − . . Our work alsoindicates that evolution in the number density of star form-ing galaxies is not enough to explain the observed redshiftevolution of XLFs, and that some additional luminosity evo-lution is preferred.It is possible to compute the likelihood L ( A, b ) =exp[ − . χ ] by fitting to the observed XLFs for any com-bination of A and b describing the redshift evolution of c X (see Eq. 6). However, we found that this formal fit is dom-inated by the two lowest luminosity data points at z . c X with the XLFs. Instead,we show in Figure 6 an example of a model where c X in-creases both with redshift and at high SFR: this model with A = 1 . × erg s − [ M ⊙ yr − ] − , b = 1 .
0, and c X → c X at SFR > M ⊙ yr − fits the observed XLFs much better.The value b = 1 . c X by a factor of 3at SFR > M ⊙ yr − is also consistent with the soft XRB(see the top right panel of Fig. 4). However, boosting c X by a factor of 3 at SFR > M ⊙ yr − appears inconsistentwith direct constraints on the L X -SFR relation. In Figure 6we used c X = 4 . z ) × erg s − [ M ⊙ yr − ] − atSFR > ∼ M ⊙ yr − which translates to c X = 8 . × ergs − [ M ⊙ yr − ] − at z = 1, while Mineo et al. (2011b) found c X = 3 . × erg s − [ M ⊙ yr − ] − for their sample ofunresolved, high SFR, sources at z ∼ . − .
2. However,the sample of high-SFR, high-z galaxies that was studied byMineo et al. (2011b) is rather limited, and if the dispersionaround this mean quantity is also 0.4 dex, then our modelmay be consistent with the observed dispersion around thisvalue out to z ∼ Figure 5.
The observed X-Ray luminosity functions (XLFs) ofgalaxies from Tzanavaris & Georgantopoulos (2008, data points )and Georgakakis et al (2006, the red dotted line shows their best-fit Schechter function) are compared to our fiducial model c X = A (1 + z ) b with A = 2 . × erg s − [ M ⊙ yr − ] − , and b = 0 . z . ∼ −
3. The z . c x to belower by ∼ . L X , . − . > ∼ erg s − . This suggeststhat either b >
0, or that c X increases at high ψ (see text). Figure 6.
Same as Figure 5, except that here we improved thefit to the observations by forcing c X to evolve with redshift andstar formation rate. The model now assumes c X = A (1 + z ) b with A = 1 . × erg s − [ M ⊙ yr − ] − , and b = 1 .
0. Furthermore,we boosted c X by a factor of ∼ > ∼ M ⊙ yr − . This plotillustrates that the XLFs favor an increase of c X with redshiftand/or at larger SFR. Observations have established that a correlation exists be-tween the star formation rate of galaxies and their X-ray lu-minosity (measured over the range E=0.5-8 keV, e.g. Ranalliet al. 2003, Grimm et al. 2003, Gilfanov et al. 2004, Lehmeret al. 2010, Mineo et al. 2010). This ‘ L X -SFR relation’ en-codes a wealth of information on various astrophysical pro- c (cid:13) , 1–12 RB Constraints on the SFR-L X Relation cesses, and strongly affects the thermal evolution of the in-tergalactic medium during the early stages of the epoch ofreionization. Existing observations have only been able toprobe this relation in nearby galaxies, and while theoreti-cally there are good reasons to suspect that c x ≡ L X / SFRincreases towards higher redshifts, observational constraintsare virtually non-existent.In this paper, we have investigated whether it is possi-ble to put any constraints on the evolution of c x with eitherredshift and/or towards high/low star formation rates thanprobed by existing observations of individual galaxies. Aspart of our analysis, we have computed that the observed‘local’ relation, when combined with the most observationalconstraints on the redshift-evolution of the star formationrate density of our Universe, implies that star forming galax-ies contribute ∼ −
15% of both the soft and ∼ −
20% ofthe hard X-ray backgrounds (see § < Γ <
3. The ob-served Γ of ULX spectra in the Chandra bands is describedby a distribution with a mean of h Γ i = 1 .
7, and a mode ofΓ pk ∼ . . − . ∼ − n ( ψ, z )], andcomputed what the contribution of ‘X-ray faint’ star forminggalaxies to the soft X-ray background (SXB, correspondingto 1-2 keV in the observed frame) is. We found that galaxieswhose individual observed flux is s s max = 2 . × − ergs − cm − between 1-2 keV, i.e. more than an order of mag-nitude fainter than the detection threshold in the ChandraDeep Field-North (see § < ∼ § c X . When we parametrize the redshift evolution as c X = A (1 + z ) b , we found that the unresolved SXB requiresthat b < ∼ . σ ). We have also ruled out models in which c X is boosted by a factor of f > ∼ − z > − . − . M ⊙ yr − , as they overproduce the unresolvedSXB (see left panels of Fig. 4). We have found indicationsin the observed X-ray luminosity functions (XLFs) of starforming galaxies that c X is increasing towards higher red-shifts and/or higher star formation rates , but caution thatthis may indicate the presence of unidentified low luminosityAGN. The unresolved SXB allows for larger changes in c X at large values for SFR (see the top right panel of Fig. 4).Finally, we also found that the SXB puts weak constraints on possible strong evolution ( f ∼ z > lower right panel of Fig. 4) .There are many other undetected candidate sourceswhich likely also contribute to the unresolved SXB. Theseare briefly summarized below (see Dijkstra et al. 2004, for amore detailed summary): • Observed AGN account for ∼
80% of the SXB. It wouldbe highly unlikely that fainter AGN– i.e. those are too faintto be detected as individual X-ray sources–do not provide asignificant contribution to the unresolved SXB. • Our attention has focused on HMXBs, but low massX-Ray binaries (LMXBs)-in which the primary has a mass < ∼ M ⊙ , dominate the X-ray luminosity of galaxies forwhich the specific star formation is sSFR < ∼ − yr − (Gilfanov et al. 2004; Lehmer et al. 2010). LMXBs give riseto a correlation between X-ray luminosity and total stellarmass, M ∗ , which is L X , LMXB ∼ × M ∗ erg s − (Gilfanov2004; Lehmer et al. 2010). In Appendix C we repeat the cal-culation of § n ( ψ, z ) with the stellar mass function n ( M ∗ , z ), and appro-priately replace P (log L X | ψ ) with P (log L X | M ∗ ). We foundthat faint ‘quiescent’ galaxies contribute about an order ofmagnitude less to the SXB than faint star-forming galaxies. • Thomson scattering of X-rays emitted predominantlyby high-redshift sources can cause 1.0-1.7% of the SXB to bein a truly diffuse form (So ltan 2003). Similarly, intergalacticdust could scatter X-rays by small angles into diffuse halosthat are too faint to be detected individually (Petric et al.2006; Dijkstra & Loeb 2009). • Wu & Xue (2001) computed that clusters and groups ofgalaxies possibly contribute as much as ∼
10% of the totalSXB. • A (hypothetical) population of ‘miniquasars’ poweredby intermediate mass black holes may have contributedto ionizing and heating the IGM (Madau et al. 2004;Ricotti & Ostriker 2004). These miniquasars would emithard X-ray photons that could contribute significantly to thesoft and hard X-ray backgrounds (Ricotti & Ostriker 2004;Dijkstra et al. 2004).The likely existence of these additional contributors tothe unresolved SXB implies that our constraints are conser-vative, and that actual limits on the redshift evolution of c X should be tighter.After our paper was submitted, Cowie et al. (2011) com-pared the average X-ray fluxes (obtained by a stacking anal-ysis) and restframe UV flux densities of sources with knownredshifts in the 4 Ms exposure of the CDF-S field. Cowieet al. (2011) showed that this ratio –after an extinction Treister et al. (2011) stacked 197 HST detected candidate z ∼ L X , − = 6 . × erg s − , which theyassociate with obscured AGN. We can use this detection to placean upper limit on c X at z = 6. The mean star formation rate–averaged over the UV-luminosity function in the range − . 0, and not corrected for dust– is ∼ M ⊙ yr − .The stacked X-ray detection therefore puts an upper limit on theboost f L X , − / . c X ∼ . We verified that such a boostat z jump . (cid:13) , 1–12 Dijkstra et al. correction– was consistent with the local L X –SFR relationup to z ∼ 4. The stacking of many source allowed Cowie etal. (2011) to probe down to s ∼ s max / × − erg s − cm − , which translates to a luminosity of L X ∼ . − × erg s − at z = 1 − 3. Cowie et al. (2011) therefore probe theredshift-evolution of c X at SFR > ∼ − M ⊙ yr − at theseredshifts (depending on z and c X ). For comparison, we haveshown that the SXB allows for constraints at lower SFR,but that the SXB becomes less sensitive to changes in c X at z > ∼ 2. Our results in combination with those of Cowie etal. (2011) thus provide stronger constraints on the allowedredshift evolution of c X . Interestingly, a non-evolution in c X with redshift appears at odds with the observed redshift evo-lution in the XLFs (unless these are contaminated by lowluminosity AGN, see above).Constraints on the redshift evolution of the L X –SFRrelation will be helpful in pinning down the astrophysics thatis driving the L X -SFR relation, and may eventually give usnew insights into the X-ray emissivity of the first galaxieswhich plays a crucial role in determining the thermal historyIGM during the dark ages (Mirabel et al. 2011). Acknowledgements We thank David Schiminovich,Tassos Fragos, and Stefano Mineo for helpful discussions.MD thanks Harvard’s Institute for Theory and Compu-tation (ITC) for its kind hospitality. This work was sup-ported in part by NSF grant AST-0907890 and NASA grantsNNX08AL43G and NNA09DB30A (for A.L.). REFERENCES Alexander, D. M., et al. 2003, AJ, 126, 539Alvarez, M. 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A., & Lahav, O. 1996, MNRAS, 280, 469Tzanavaris, P., & Georgantopoulos, I. 2008, A&A, 480, 663 c (cid:13) , 1–12 RB Constraints on the SFR-L X Relation Figure A1. This plot shows the star formation rate density (co-moving) in the Universe, ˙ ρ ∗ ( z ). The black solid line shows ˙ ρ ∗ ( z )that has been derived by Hopkins & Beacom (2006). The datapoint at z = 0 represents the more recent z = 0 estimate by Both-well et al. (2011). The blue dashed line ( red dotted line ) shows ourmodel in which we evolve ψ ∗ (Φ ∗ ) in our adopted star formationrate function with redshift, while keeping Φ ∗ ( ψ ∗ ) fixed. Bothmodels clearly reproduce the ‘observed’ redshift evolution of thestar formation rate density. Venkatesan, A., Giroux, M. L., & Shull, J. M. 2001, ApJ,563, 1Willott, C. J. 2011, arXiv:1110.4118Wise, J. H., & Cen, R. 2009, ApJ, 693, 984Wu, X.-P., & Xue, Y.-J. 2001, ApJ, 560, 544Zheng, Z., Cen, R., Trac, H., & Miralda-Escud´e, J. 2010,ApJ, 716, 574 APPENDIX A: ASSUMED REDSHIFTEVOLUTION OF THE STAR FORMATIONRATE FUNCTIONA1 The Integrated Star Formation Rate Density In our paper, we needed to assume the redshift evolution ofthe star formation rate function ( dn/dψ ). We studied twomodels: • In our first model, we evolved ψ ∗ ( z ) to match the ob-served star formation rate density, but kept Φ ∗ fixed. Wefound that the following redshift evolution of ψ ∗ ( z ) providesa decent fit to observations: ψ ∗ ( z ) M ⊙ yr − = . z ) . z < z ;9 . z ) . z z < z . z ) . (cid:16) z z (cid:17) . z > z , (A1)where z = 1 . z = 3 . 35. The resulting integratedstar formation rate density is shown as the blue dashedline in Figure A1, which should be compared to that de-rived by Hopkins & Beacom (2006), which is shown as the black solid line . Both curves clearly agree. Note that ouradopted star formation rate function–which was compiledfrom the most recent data–results in a larger star formationrate density at z = 0 ( ˙ ρ ∗ = 0 . M ⊙ yr − Mpc − , indicated by the black filled circle at z = 0), than that inferred byHopkins & Beacom (2006), which is ˙ ρ BH06 ( z = 0) = ah =0 . M ⊙ yr − Mpc − . • In our second model, we evolved Φ ∗ ( z ) to match theobserved star formation rate density, but kept ψ ∗ fixed. Weassumed the redshift evolution of Φ ∗ ( z ) wasΦ ∗ ( z ) = Φ ∗ ( z = 0) × ˙ ρ HB06 ( z )˙ ρ HB06 ( z = 0) g ( z ) , (A2)where the function g ( z ) ≡ + (1 + z ) − compensates forthe fact that the faint-end slope of the star formation ratefunction, α , changes with redshift in our model. The result-ing integrated star formation rate density is shown as the red dotted line in Figure A1. APPENDIX B: DERIVATION OF EQ 1 Eq 1 plays a central role in our analysis. Here, we pro-vide more details on its origin. The total observed flux dS from a proper (i.e physical) cosmological volume ele-ment dV p is dS ( z ) = (1 + z ) dV p ˙ ρ ∗ L ( z, Γ) / πd ( z ). Thecosmological proper volume element dV p can be written as dV p = cH dz (1+ z ) E ( z ) dA p . We substitute dA p = d ( z ) d Ω,where d A ( z ) denotes the angular diameter distance to red-shift z . We finally get for the differential flux per sterradian dS ( z ) d Ω = cH d ( z ) ˙ ρ ∗ ( z )(1 + z ) L (Γ , z )(1 + z ) E ( z )4 πd ( z ) = (B1) c πH ˙ ρ ∗ ( z ) L (Γ , z ) dz E ( z )(1 + z ) , where we used that d A ( z ) = (1 + z ) − d L ( z ). When we inte-grate over redshift and solid angle ∆Ω, we arrive at Eq 1. APPENDIX C: CONTRIBUTION OFQUIESCENT GALAXIES TO THE SXB To compute the contribution of low mass X-ray binaries tothe SXB we only need to modify Eq 3 in two ways: (i) we re-place the star formation rate function with the observed stel-lar mass function n ( M ∗ , z ), and (ii) we replace P (log L X | ψ )with P (log L X | M ∗ ). The goal of this appendix is to providemore details of the calculation, and to show that the conclu-sion that LMXBs contribute about an order of magnitudeless to the SXB than HMXBs is robust to uncertainties inthe modeling.Observed stellar mass functions can be describedby Schechter functions, and observations have con-strained the Schechter parameters out to z ∼ red dashed lines ),and Mortlock et al. (2011, black solid lines ). Both calcula-tions agree well.We assume that P (log L X | M ∗ ) is given by a lognormaldistribution, where h L X i ≡ C X M ∗ . Here, C X = 8 . ± . × erg s − M − ⊙ (Gilfanov et al. 2004). The scatter in thisrelation is not given, and for simplicity we have adopted c (cid:13) , 1–12 Dijkstra et al. Figure C1. Same as Figure 2, but for low mass X-ray binaries (LMXB), for which the cumulative luminosity scales linearly with thetotal stellar mass. We integrate the stellar mass functions down to some minimum mass M min . The upper left panel shows S X as afunction of M min . Another difference with Figure 2 is that we do not extend our calculation beyond z max = 4 . black solid lines [ red dashed lines ] show S X if we adopt the stellar mass functionsfrom Mortlock et al. (2011) [P´erez-Gonz´alez et al. 2008]. We find that typically, the contribution from LMXBs to the SXB lies about anorder of magnitude below that of HMXBs. σ = 0 . 4, but note that our results can simply be rescaledby a factor of exp (cid:0) ln σ − σ ] (cid:1) to obtain predictions forany σ . The last difference with the calculation presented inthe main paper is that L X , LMXB is measured in the restframeE= 2 − 10 keV band (Lehmer et al. 2010).Figure C1 presents results from our calculations in away that is identical to Figure 2 of the main paper. The maindifferences are: ( i ) the upper left panel shows S X as a functionof minimum stellar mass (instead of minimum star formationrate), and ( ii ) our calculations extend only out to z max =4 . 0, as the observed stellar mass functions become uncertainthere. The general trends in this figure are similar to thosein Figure 2, except the dependence of S X on Γ. This differentdependence results from the fact that L X , LMXB is measuredin the 2-10 keV band (compared to 0.5-8.0 keV for HMXBs)which introduces different K-corrections. Generally, we findthat LMXBs produce S X < ∼ × − erg s − cm − deg − ,which is ∼ 10% of the amount contributed by HMXBs. c (cid:13)000